given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim…

given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} h(x)=1 \\)\n\\( \\lim _{x \\rightarrow a} p(x)=\\infty \\)\n\\( \\lim _{x \\rightarrow a} q(x)=\\infty \\),\nevaluate if the following limits are not indeterminate forms. (if a limit is indeterminate, enter indeterminate.)\n(a) \\( \\lim _{x \\rightarrow a}f(x)-p(x) \\)\n(b) \\( \\lim _{x \\rightarrow a}p(x)-q(x) \\)\n(c) \\( \\lim _{x \\rightarrow a}p(x)+q(x) \\)

given that\n\\( \\lim _{x \\rightarrow a} f(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} g(x)=0 \\)\n\\( \\lim _{x \\rightarrow a} h(x)=1 \\)\n\\( \\lim _{x \\rightarrow a} p(x)=\\infty \\)\n\\( \\lim _{x \\rightarrow a} q(x)=\\infty \\),\nevaluate if the following limits are not indeterminate forms. (if a limit is indeterminate, enter indeterminate.)\n(a) \\( \\lim _{x \\rightarrow a}f(x)-p(x) \\)\n(b) \\( \\lim _{x \\rightarrow a}p(x)-q(x) \\)\n(c) \\( \\lim _{x \\rightarrow a}p(x)+q(x) \\)

Answer

Explanation:

Step1: Analyze (\lim_{x\rightarrow a}[f(x)-p(x)])

We know that (\lim_{x\rightarrow a}f(x) = 0) and (\lim_{x\rightarrow a}p(x)=\infty). Using the property of limits (\lim_{x\rightarrow a}[f(x)-p(x)]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow a}p(x)) Substitute the values: (0-\infty=-\infty)

Step2: Analyze (\lim_{x\rightarrow a}[p(x)-q(x)])

Since (\lim_{x\rightarrow a}p(x)=\infty) and (\lim_{x\rightarrow a}q(x)=\infty). The form (\infty-\infty) is an indeterminate form.

Step3: Analyze (\lim_{x\rightarrow a}[p(x)+q(x)])

Given (\lim_{x\rightarrow a}p(x)=\infty) and (\lim_{x\rightarrow a}q(x)=\infty) Using the property of limits (\lim_{x\rightarrow a}[p(x)+q(x)]=\lim_{x\rightarrow a}p(x)+\lim_{x\rightarrow a}q(x)) Substitute the values: (\infty+\infty=\infty)

Answer:

(a) (-\infty) (b) INDETERMINATE (c) (\infty)